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an introduction to management science

Questions and Answers of An Introduction to Management Science

=V1. The Solver has difficulty solving nonlinear programming problems with certain properties. List three of these properties.
=3. How does mutation help Evolutionary Solver?
=4. What are two advantages that Evolutionary Solver has over other solving methods in Solver for solving difficult nonlinear programming problems?
=5. What are three disadvantages of Evolutionary Solver compared to the other solving methods in Solver?
=V8.1. The J. P. Atkins Company will soon be introducing a new product. Estimates have been made of the monthly profit that would be generated by this product for each of four alternative values of
=For positive values of a andb, this will give lower demand when the price is higher. However, a nonlinear demand function usually can provide a better fit to the data. For example, one such
=. For positive values of a and negative values ofb, this also will give lower demand when the price is higher. Graph the above data and use the Add Trendline feature of Excel to find the constant
=b. For this part, assume that the airline charges a single price to all customers. Using the demand function for total demand determined in parta, formulate and solve a nonlinear programming model
=c. Now assume that the airline charges separate prices for midweek and Saturday night stay tickets. Using the two demand functions for midweek and Saturday night stay tickets determined in parta,
=d. How much extra profit can the airline achieve by charging higher prices for midweek tickets than for Saturday night stay tickets?
=a. Draw a profit graph for this product by plotting the profits for the four production rates and then drawing a smooth curve through the four points by hand. (Start the graph with a profit of 0 at
=b. Does the proportionality assumption of linear programming seem to be satisfied reasonably well for this product?
=c. To the extent that profit is not strictly proportional to the production rate, does this product have decreasing marginal returns, increasing marginal returns, or neither?
=E*d. Use Excel’s curve fitting method to (1) obtain a nonlinear formula with a quadratic form for the profit graph and then (2) construct the graph.
=8.2. Consider the following three cases for how the profit from an activity varies with the level of the activity.
=a. For each case, draw the profit graph by plotting the profits for the various levels of the activity and then drawing a smooth curve through the points by hand.
=b. For each case, indicate whether the activity has decreasing marginal returns, increasing marginal returns, or neither.
=c. How would your answers in part b change if the graphs plotted in part a were cost graphs instead of profit graphs?
=E*d. For each case, use Excel’s curve fitting method to(1) obtain a nonlinear formula with a quadratic form for the profit graph and then (2) construct the graph.For any case where the activity
=8.3. The Chiplet Corporation is about to launch the production and marketing of a new microchip that is more powerful than anything that is currently on the market. Not surprisingly, the
=Va. Draw a profit graph for this microchip by plotting the profits for the various sales levels and then drawing a smooth curve that passes through (or very near) these points.
=b. Does the microchip have decreasing marginal returns, increasing marginal returns, or neither?
=E*c. Use Excel’s curve fitting method to (1) obtain a nonlinear formula with a quadratic form (a polynomial of order 2) for the profit graph and then (2) construct the graph.
=E*d. Repeat part c when using the Excel option of a polynomial of order 3 instead of order 2.
=e. Which of the Excel options used in parts c and d does a better job of fitting the profit graph to the data?
=8.4. The following table shows the estimated daily profit from a new product for several of the alternative choices for the production rate.Because the profit goes up less than proportionally with
=a. One such approximation is P $100R $5R2.How closely does this nonlinear function approximate the five values of P given in the table?
=b. Repeat part a for the approximation, P $104R$6R2.
=c. Which of these two nonlinear functions provides the better fit to all the data?
=E*d. Use Excel’s curve fitting method to (1) obtain a nonlinear formula with a quadratic form for the profit graph and then (2) construct the graph.
=E*8.6. Reconsider the portfolio selection example, including its spreadsheet model in Figure 8.13, given in Section 8.2. Note in Table 8.2 that stock 2 has the highest expected return and stock 3
=c. Use the Solver Table to systematically try all the percentages at 5% intervals from 0% to 50%.
=8.7.* A stockbroker, Richard Smith, has just received a call from his most important client, Ann Hardy. Ann has $50,000 to invest and wants to use it to purchase two stocks. Stock 1 is a solid
=a. Without yet assigning a specific numerical value to the minimum acceptable expected profit, formulate a quadratic programming model in algebraic form for this problem.
=E*b. Display this model on a spreadsheet.
=E*c. Solve this model for four cases: Minimum acceptable expected profit $13,000, $15,000, $17,000, and $19,000.
=d. Ann was a statistics major in college and so understands well that the expected return and risk in this model represent estimates of the mean and standard deviation of the probability
=8.8. Reconsider the portfolio selection example given in Section 8.2. A fourth stock (stock 4) now has been found that gives a good balance between expected return and risk. Using the same units as
=a. Still using a minimum acceptable expected return of 18%, formulate the revised quadratic programming model in algebraic form for this problem.
=E*c. Develop a revision of the Solver Table shown in Figure 8.14 for this revised problem.
=8.9. The management of the Albert Hanson Company is trying to determine the best product mix for two new products.Because these products would share the same production facilities, the total
=a. Formulate a quadratic programming model in algebraic form for determining the product mix that maximizes the total profit per hour.
=E*b. Formulate and solve this model on a spreadsheet.8.10. The B. J. Jensen Company specializes in the production of power saws and power drills for home use. Sales are relatively stable throughout
=B. J. Jensen, Jr., the current president of the company, is overseeing the production plans being made for the upcoming November. He has obtained the data at the top of the next page.However, Mr.
=a. Draw the profit graph for each of these two products.
=E*b. Use separable programming to formulate a linear programming model on a spreadsheet for this problem. Then solve the model. What does this say about how many power saws and how many power
=8.11.* The Dorwyn Company has two new products (special kinds of doors and windows) that will compete with the two new products for the Wyndor Glass Co. (described in Section 2.1).Using units of
=E*a. Formulate and solve this nonlinear programming model on a spreadsheet.
=b. Construct tables to show the profit data for each product when the production rate is 0, 1, 2, 3.
=c. Draw a figure that plots the weekly profit points for each product when the production rate is 0, 1, 2, 3.Connect the pairs of consecutive points with(dashed) line segments.
=E*d. Use separable programming based on this figure to formulate an approximate linear programming model Profit 4D 6W D3 2W2 D 0 W 0 5D 2W 14 D 3W 8 Maximize Profit 4D 6W on a spreadsheet for this
=e. Compare the solution based on a separable programming approximation in part d with the solution obtained in part a for the exact nonlinear programming model.
=8.12. The MFG Corporation is planning to produce and market three different products. Let x1, x2, and x3 denote the number of units of the three respective products to be produced. The preliminary
=a. Plot the profit graph for each of the three products.
=E*b. Use separable programming to formulate a linear programming model on a spreadsheet for this problem. Then solve the model. What is the resulting recommendation to management about the values
=8.13. Suppose that separable programming has been applied to a certain problem (the “original problem”) to convert it to the following equivalent linear programming model in algebraic
=VWhat was the mathematical model for the original problem?Answer this by plotting the profit graph for each of the original activities and then writing the constraints for the original problem in
=8.14. Jim Matthews, vice president for marketing of the J. R. Nickel Company, is planning advertising campaigns for two unrelated products. These two campaigns need to use some of the same
=a. Construct tables to show the profit data for each product when the level of its advertising campaign is x1 0, 1, 2, 2.5, 3, 4, 5 (for the first product) or x2 0, 1, 2, 3, 3.5, 4, 5 (for the
=c. On the profit graph for the first product, draw an approximation of this profit graph by inserting a dashed-line segment between the profit at x1 0 and x1 2, between the profit at x1 2 and x1 4,
=E*d. Use separable programming with the approximation of the profit graphs obtained in part c to formulate an approximate linear programming model on a spreadsheet for Jim Matthews’s problem. Then
=E*e. Repeat parts c and d except using x1 0, 2, 2.5, 3, 5 and x2 0, 3, 3.5, 4, 5 for the approximations of the profit graphs in partc. (These particular approximations actually lead to the exact
=E*f. Use Excel and its Solver to formulate and solve the original nonlinear programming model directly. Compare with the answers obtained after completing part e.g. Use calculus to find the value
=E*8.15. Consider the following nonlinear programming problem.subject to
=a. Formulate this problem in a spreadsheet and then use the Solver Table to solve this problem with the following starting points: x 0, 1, 2, 3, 4, and 5.Include the value of x and the profit as
=b. Use Evolutionary Solver to solve this problem.
=E*8.16. Consider the following nonlinear programming problem.subject to
=a. Formulate this problem in a spreadsheet and then use the Solver Table to solve this problem with the following starting points: x 0, 1, 2, 3, 4, and 5.Include the value of x and the profit as
=b. Use Evolutionary Solver to solve this problem.
=E*8.17. Because of population growth, the state of Washington has been given an additional seat in the House of Representatives, making a total of ten. The state legislature, which is currently
=8.18. Reconsider the portfolio optimization problem considered in Section 8.5, where the goal was to select the portfolio that beat the market for the largest number of quarters over the last six
=E*a. Using the naive solution (20 percent in each stock)as a starting point, apply Evolutionary Solver to Democrat Republican City (Thousands) (Thousands)1 152 62 2 81 59 3 75 83 4 34 52 5 62 87 6
=c. Comment on the results from parts a and b.E*8.19. Reconsider the portfolio optimization problem considered in Section 8.5, where the goal was to select the portfolio that beat the market for the
=a. Use Evolutionary Solver to instead find a portfolio that did not lose money in the largest number of quarters.
=b. Use Evolutionary Solver to instead find a portfolio that yielded a return of at least 10 percent for the largest number of quarters.8.20. Reconsider the Wyndor Glass Co. problem introduced in
=E*a. Solve this problem using the standard Solver.
=E*b. Starting with an initial solution of producing 0 doors and 0 windows, solve this problem using Evolutionary Solver.
=c. Comment on the performance of the two approaches.
=a. For each of the three advertising media, draw a graph of the number of sales versus the number of advertisements by plotting the sales for the five points provided by Sid Jackowitz and then
=a portion of the available outlets.)
=b. For each of the advertising media, use Excel’s curve fitting method to (1) obtain a nonlinear formula for the sales graph and then (2) construct the graph. In each case, try three Excel options
=c. Using your results from partb, write an expression for the total profit (as defined by Claire) in terms of the number of advertisements of each type.
=d. Using your result from partc, revise the spreadsheet model in Figure 3.7 (available on the CD-ROM) so that it maximizes total profit instead of the total number of exposures, and then solve.
=e. Use the sales tables provided by Sid Jackowitz to apply separable programming to this problem when maximizing total profit.
=f. Compare your results in parts d and e with those in Figure 3.7 and then give your recommendation (with a brief explanation)for the best advertising mix. Do you feel it was worthwhile to introduce
=a. At first, Lydia wants to ignore the risk of all the investments. Given this strategy, what is her optimal investment portfolio; that is, what fraction of her money should she invest in each of
=b. Lydia decides that she doesn’t want to invest more than 40 percent in any individual stock. While still ignoring risk, what is her new optimal investment portfolio? What is the total risk of
=c. Now Lydia wants to take into account the risk of her investment opportunities. For use in the following parts, formulate a quadratic programming model that will minimize her risk(measured by the
=d. Lydia wants to ensure that she receives an expected return of at least 35 percent. She wants to reach this goal at minimum risk. What investment portfolio allows her to do that?
=e. What is the minimum risk Lydia can achieve if she wants an expected return of at least 25 percent? Of at least 40 percent?
=f. Do you see any problems or disadvantages with Lydia’s approach to her investment strategy?
=a. Formulate a separable programming model to be used in the following parts.
=b. What is the optimal investment strategy for Charles?
=c. What is fundamentally wrong with the advice Charles got from the investment advisor at the bank?
=d. Now that Charles is considering investing in the certificate of deposit, what is his optimal investment strategy?
=e. What would his optimal investment strategy for the fifth, sixth, and seventh years have been if he had originally invested DM 50,000?
=f. Charles and his fiancée have been planning to get married after his first year in business school. However, Charles learns that for married couples, the tax-free amount of interest earnings
=g. Due to a recession in Germany, interest rates are low and are expected to remain low. However, since the American economy is booming, interest rates are expected to rise in the United States. A
=1. Identify the kind of decision-making environment for which decision analysis is needed.
=2. Describe the logical way in which decision analysis organizes a problem.

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